Difference between revisions of "Quotient group"

Latest revision as of 20:15, 23 October 2016

Definition

Let [ilmath](G,\times)[/ilmath] be a group. Let [ilmath]H\subseteq G[/ilmath] be a normal subgroup of [ilmath]G[/ilmath]. Then:

• the cosets of [ilmath]H[/ilmath] in [ilmath]G[/ilmath] form a group whose operation is the group operation on subsets^{[Note 1]}

We denote this new group [ilmath]\frac{G}{H} [/ilmath] or [ilmath]G/H[/ilmath].

With it we also get a group homomorphism called the canonical projection of the quotient group:

• [ilmath]\pi:G\rightarrow\frac{G}{H} [/ilmath] given by [ilmath]\pi:g\rightarrow [g][/ilmath] where [ilmath][g][/ilmath] denotes the coset containing [ilmath]g[/ilmath].

Caution:This requires some work, why must the cosets of [ilmath]H[/ilmath] in [ilmath]G[/ilmath] partition [ilmath]G[/ilmath]?

See also

• Normal subgroup
  • Every subgroup of an abelian group is normal

Notes

1. ↑ Let [ilmath]A\subseteq G[/ilmath] be an arbitrary subset of a group [ilmath](G,\times)[/ilmath] and let [ilmath]g\in G[/ilmath] be given. Then:
   • [ilmath]g\times A:=\{g\times a\ \vert\ a\in A\}[/ilmath]
   By itself this isn't very interesting.

References